A word for people who work under a manager

Solution 1:

Staff

a group of officers assisting an officer in command of an army formation or administration headquarters. (OED)

Although the dictionary appears to restrict this particular usage to the military, it is often used in the exact context you describe.

Solution 2:

Where I used to work, we called the people who reported to a manager his/her reports. This word does not have any of the negative connotations words like subordinates or underlings carry.

Oxford Dictionaries Online lists this as the meaning of the word and also gives an example.

Report
noun
An employee who reports to another employee
'And, I have been a better, more consistent mentor/teacher to my direct reports.'

Solution 3:

Subordinates

adjective 1. placed in or belonging to a lower order or rank.

Sounds too negative? How about Team?